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Problem 150

Number theory Difficulty 1.4 Multiple choice CEMC Gauss (Grade 8) · Canada · 2022

The 26 letters of the English alphabet are listed in an infinite,
repeating loop:

$ABCDEFGHIJKLMNOPQRSTUVWXYZABC
$\dots\$$

What is the 258th letter in this sequence?

Pick one

Official solution

Since 42 is an even number, then 2 is a factor of 42.

Since $42=2\times21$ and $21=3\times7$ , then $42=2\times3\times7$ .

Each of $2,3$ and 7 is a prime number
and each is a factor of 42.

Thus, the sum of the prime factors of 42 is $2+3+7=12$ .

(We note that $1,6,14,21$ , and 42 are
also factors of 42, however they are not prime numbers.)

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.